Solving Linear Equations and Linear Inequalities: Basic Example

Key Concepts

• Objective: Simplify the given inequality, (3l - 6 \geq 8), to find the solutions for (l).
• Steps to Solve Inequalities:
  • Eliminate Constants: Start by removing constants from one side by performing the inverse operation.
  • Balance the Inequality: Always perform the same operation on both sides to maintain equality.
  • Solve for the Variable: Isolate the variable to find its possible values.

Step-by-Step Solution

1. Eliminate (-6) by Addition

   • Add 6 to both sides of the inequality: [3l - 6 + 6 \geq 8 + 6]
   • Simplified to: [3l \geq 14]

2. Divide by Coefficient

   • Divide each side by 3 (the coefficient of (l)): [\frac{3l}{3} \geq \frac{14}{3}]
   • Resulting in: [l \geq \frac{14}{3}]

Important Rules

• Maintain Inequality: When you add/subtract/multiply/divide by a positive number, the direction of the inequality remains the same.
• Change Inequality Direction: When multiplying or dividing by a negative number, reverse the direction of the inequality.

Applications

• Understanding Inequalities: This process demonstrates how to balance and solve inequalities, a key skill in algebra.
• SAT Preparation: These types of problems are foundational in standardized tests like the SAT.

Additional Tips

• Order of Operations (PEMDAS): While solving expressions, follow the order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Reverse Operations (SADMEP): For solving equations and inequalities, you apply operations in reverse order when isolating the variable: Subtraction/Add, Divide/Multiply, Exponents, Parentheses.

Instructor's Explanation

• Conceptual Focus: Understand why each step is necessary to maintain the equality/inequality.
• Questions to Consider:
  • Why do both sides need the same operation?
  • What happens if you divide by a negative number?

Example Problem: Solving (3l - 6 \geq 8) demonstrates applying these principles effectively.